Rigidity and Realizability for Tropical Curves in Dimension 3

2025/02/23 by Hicks, Jeff
#14T20 #53D40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2502.16582

Abstract

We present an unobstructedness criterion for Lagrangian threefolds L⊂ X^A using the H₁(L)-class associated with the boundary of a pseudoholomorphic disk. As an application, let X^A→ Q be a Lagrangian torus fibration whose base Q is a tropical abelian threefold. Given V⊂ Q a rigid tropical curve with a pair-of-pants decomposition, we prove that the Lagrangian lift L_V⊂ X^A is unobstructed. Provided that an appropriate homological mirror symmetry statement holds, this implies the existence of a realization Y_V in the mirror abelian threefold X^B→ Q.

Related

• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Examples of tropical-to-Lagrangian correspondence
• Obstructions to lifting tropical curves in surfaces in 3-space
• The Realizability of Curves in a Tropical Plane
• Lagrangian cobordism and tropical curves